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Value sets of bivariate folding polynomials over finite fields
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Date
2018-11-01
Author
Küçüksakallı, Ömer
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We find the cardinality of the value sets of polynomial maps associated with simple complex Lie algebras B-2 and G(2) over finite fields. We achieve this by using a characterization of their fixed points in terms of sums of roots of unity.
Subject Keywords
Lie algebra
,
Weyl group
,
Fixed point
,
Permutation
URI
https://hdl.handle.net/11511/32985
Journal
FINITE FIELDS AND THEIR APPLICATIONS
DOI
https://doi.org/10.1016/j.ffa.2018.05.008
Collections
Department of Mathematics, Article
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Value sets of folding polynomials over finite fields
Küçüksakallı, Ömer (2019-01-01)
Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Bivariate polynomial mappings associated with simple complex Lie algebras
Küçüksakallı, Ömer (2016-11-01)
There are three families of bivariate polynomial maps associated with the rank-2 simple complex Lie algebras A(2), B-2 congruent to C-2 and G(2). It is known that the bivariate polynomial map associated with A(2) induces a permutation of F-q(2) if and only if gcd(k, q(3) - 1) = I. for s = 1, 2, 3. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.
Value sets of Lattes maps over finite fields
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Küçüksakallı, Ömer (2015-11-01)
We determine the cardinality of the value sets of bivariate Chebyshev maps over finite fields. We achieve this using the dynamical properties of these maps and the algebraic expressions of their fixed points in terms of roots of unity.
Exceptional Lie algebra g2 and its representations
Kayakökü, Mehmet Mustafa; Ünal, İbrahim; Department of Mathematics (2022-9-01)
In the classification of complex simple Lie algebras, there are five of them whose Dynkin diagrams are of exceptional type. The Lie algebra g_2 has the smallest dimension among these exceptional Lie algebras and together with its corresponding Lie group G_2, it plays an important role in differential geometry, mathematical physics, and modern string theory. In this thesis after a general introduction to Lie algebras, we show the classification of complex simple ones. Afterward, we give several constructions...
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Ö. Küçüksakallı, “Value sets of bivariate folding polynomials over finite fields,”
FINITE FIELDS AND THEIR APPLICATIONS
, pp. 253–272, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32985.